Betting Strategy and Ⓜodel Validation - Part II

Betting Model Analysis on Sportsbook Consultancy Firm A

®γσ, Eng Lian Hu 白戸則道®

2016-09-16

Abstract

This is an academic research by apply R statistics analysis to an agency A of an existing betting consultancy firm A. According to the Dixon and Pope (2004)1 Kindly refer to 24th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References, due to business confidential and privacy I am also using agency A and firm A in this paper. The purpose of the anaysis is measure the staking model of the firm A. For more sample which using R for Soccer Betting see http://rpubs.com/englianhu. Here is the references of rmarkdown and An Introduction to R Markdown. You are welcome to read the Tony Hirst (2014)2 Kindly refer to 1st paper in Reference for technical research on programming and coding portion for the paper. in 7.4 References if you are getting interest to write a data analysis on Sports-book.

1. Introduction to the Betting Stategics

2. Data

3. Summarise the Staking Model

4. Staking Ⓜodel

4.1 Basic Equation

Before we start modelling, we look at the summary of investment return rates.

table 4.1.1 : 5 x 5 : Return of annually investment summary table.3 Kindly refer to the list of colors via Dark yellow with hexadecimal color code #9b870c for plot the stylist table.

\[\Re = \sum_{i=1}^{n}\rho_{i}^{EM}/\sum_{i=1}^{n}\rho_{i}^{BK} \cdots equation 4.1.1\]

\(\Re\) is the return rates of investment. The \(\rho_i^{EM}\) is the estimated probabilities which is the calculated by firm A from match 1,2… until \(n\) matches while \(\rho_{i}^{BK}\) is the net/pure probability (real odds) offer by bookmakers after we fit the equation 4.1.2 into equation 4.1.1.

\[\rho_i = P_i^{Lay} / (P_i^{Back} + P_i^{Lay}) \cdots equation 4.1.2\]

\(P_i^{Back}\) and \(P_i^{Lay}\) is the backed and layed fair price offer by bookmakers.

We can simply apply equation above to get the value \(\Re\). From the table above we know that the EMPrice calculated by firm A invested at a threshold edge (price greater) 1.0769894, 1.1072203, 1.0781056, 1.1148426, 1.0671108 than the prices offer by bookmakers. There are some description about \(\Re\) on Dixon and Coles (1996)4 Kindly refer to 25th paper in Reference for industry knowdelege and academic research portion for the paper. under 7.4 References. The optimal value of \(\rho_{i}\) (rEMProbB) will be calculated based on bootstrapping/resampling method in section 4.3 Kelly Ⓜodel.

table 4.1.2 : 48640 x 45 : Odds price and probabilities sample table.

Above table list a part of sample odds prices and probabilities of soccer match \(i\) while \(n\) indicates the number of soccer matches. We can know the values rEMProbB, netProbB and so forth.

graph 4.1.1 : A sample graph about the relationship between the investmental probabilities -vs- bookmakers’ probabilities.

Graph above shows the probabilities calculated by firm A to back against real probabilities offered by bookmakers over 48640 soccer matches.

Now we look at the result of the soccer matches.

table 4.1.3 : 7 x 8 : Summary of betting results.

The table above summarize the stakes and return on soccer matches result. Well, below table list the handicaps placed by firm A on agency A. I list the handicap prior to test the coefficient according to the handicap in next section 4.2 Linear Ⓜodel.

table 4.1.4 : 6 x 8 : The handicap in sample data.

4.2 Linear Ⓜodel

From our understanding of staking, the covariates we need to consider should be only odds price since the handicap’s covariate has settled according to different handicap of EMOdds.

Again, I don’t pretend to know the correct Ⓜodel, here I simply apply linear model to retrieve the value of EMOdds derived from stakes. The purpose of measure the edge overcame bookmakers’ vigorish is to know the levarage of the staking activities onto 1 unit edge of odds price by firm A to agency A.

table 4.2.1 : Summary of linear models.

table 4.2.2 : Anova of linear models.

Due to the coefficient of the regression models ocuppied a long space in this article. Here I write in a shiny app which perform the summary and anova of the models. Kindly click on Report with ShinyApps to use the ShinyApp. It will take few minutes time to open the website since the dataset contain 5 years soccer data from firm A staking via agent A made it a heavily loading.

When I used to work in 188Bet and Singbet as well as AS3388, we know from the experience which is the odds price of favorite team win will be the standard reference and the draw odds will adjust a little bit while the underdog team will be ignore.

Steven Xu (2013)5 Kindly refer to 16th paper in Reference for industry knowdelege and academic research portion for the paper. has do a case study on the comparison of the efficiency of opening and closing price of NFL and College American Football Leagues and get to know the closing price is more efficient and accurate compare to opening price nowadays compare to years 1980~1990. It might be due to multi-million dollars of stakes from informed traders or smart punters to tune up the closing price to be likelihood.

In order to test the empirical clichés, I used to conduct a research thoroughly through ®γσ, Eng Lian Hu (2016)6 Kindly refer to 3rd paper in Reference for industry knowdelege and academic research portion for the paper. under 7.4 References, I completed the research on year 2010 but write the thesis in year 2016. and concludes that the opening price of Asian Handicap and also Goal Lines of 29 bookmakers are efficient than mine. However in my later ®γσ, Eng Lian Hu (2014)[Kindly refer to 4th paper in Reference for industry knowdelege and academic research portion for the paper. under 7.4 References] applied Kelly staking model where made a return of more than 30% per sesson. Meanwhile, the Dixon and Coles (1996) and Crowder, Dixon, Ledford and Robinson (2001)7 Kindly refer to 27th paper in Reference for industry knowdelege and academic research portion for the paper. under 7.4 References has built two models which compare the accuracy of home win, draw and away win. From a normal Poison model reported the home win is more accurate and therefore an add-hoc inflated parameter required in order to increase the accuracy of prediction. You are feel free to learn about the Dixon and Coles (1996) in section 4.4 Poisson Ⓜodel.

Kindly click on Report with ShinyApps to use the ShinyApp

Kindly click on Report with ShinyApps to use the ShinyApp

shinyapp 4.2.1 : Summary and anova of linear models.

Based on table 2.2.1 we know about the net bookies probabilities and EM probabilities, here I simply apply linear regression model8 You can learn from Linear Regression in R (R Tutorial 5.1 to 5.11). You can also refer to Getting Started with Mixed Effect Models in R, A very basic tutorial for performing linear mixed effects analyses and Fitting Linear Mixed-Effects Models using lme4. Otherwise you can read Linear Models with R and somemore details about regression models via Extending the Linear Model with R : Generalized Linear, Mixed Effects and Nonparametric Regression Models. Besides, What statistical analysis should I use? summarise a table for test analysis and data validation. and also anova to compare among the models.

You might select Y response variable and X explanatory variable(s) to measure your model9 Refer to Shiny height-weight example for further information about shinyapp for linear models. or existing models.

Kindly click on Report with ShinyApps to use the ShinyApp

Kindly click on Report with ShinyApps to use the ShinyApp

shinyapp 4.2.2 : Comparison of the vigorish between AH and Fixed-Odds. Source from 1x2 to Combo AH version 1.1 by William Chen (2006)

Here I simply attached with a Fixed Odds to Asian Handicap’s calculator which refer to my ex-colleague William Chen’s10 My ex-colleague and best friend in sportsbook industry which known since join sportsbook industry year 2005 —— Telebiz and later Caspo Inc. spreadsheet version 1.1 in year 2006. You can simply input the home win, draw, away win (in decimal format) as well as the overround.11 Kindly refer to my previous research to know the vigorish / overround.

Section : reverse modelling to get the EMProb prior to calculate the coefficient of the staking model. Otherwise might rearrange the order of applied Poison model here by refer to international competitions.

Galema, Plantinga and Scholtens (2008)12 You are feel free to refer to [Reference for industry knowdelege and academic research portion for the paper] in 7.4 References for further details

reminder (temporary noted for further):

draft : 
  - http://www.moneychimp.com/articles/risk/regression.htm
  - read *Galema, Plantinga and Scholtens (2008)* https://englianhu.files.wordpress.com/2016/06/the-stocks-at-stake-return-and-risk-in-socially-responsible-investment.pdf
  - reverse engineering on staking-profit linear regression model to get/retrieve EMProb value since now only get the coefficients figire of EMProb. Although incompleted soccer teams... 2ndly, reversed poison model from EMProb might is not workable on one-sided competition, need to refer to some international competition as references for incompleted dataset.

John Fingleton & Patrick Waldron (1999)13 Kindly refer to 13th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References apply Shin’s model and finally conclude suggests that bookmakers in Ireland are infinitely risk-averse and balance their books. The authors cannot distinguish between inside information and operating costs, merely concluding that combined they account for up to 3.7% of turnover while normally Asian bookmakers made less than 1% and a anonymous company has made around 2%. However the revenue or the stakes are farly more than European bookmakers.14 You can refer to my another project Analyse the Finance and Stocks Price of Bookmakers which analysis the financial report of public listed companies and also profitable products’ revenue and profit & loss of anonymous company..

They compare different versions of our model, using data from races in Ireland in 1993. The authors’ empirical results can be summarised as follows:

  • They reject the hypothesis that bookmakers behave in a risk neutral manner;
  • They cannot reject the hypothesis that they are infinitely riskaverse;
  • They estimate gross margins to be up to 4 per cent of total oncourse turnover; and
  • They estimate that 3.1 to 3.7% (by value) of all bets are placed by punters with inside information.

Here I try to test our data if there has any insider information.

4.3 Kelly Ⓜodel

From the papers Niko Marttinen (2001)15 Kindly refer to 1th paper in Reference for industry knowdelege and academic research portion for the paper. and Jeffrey Alan Logan Snyder (2013)16 Kindly refer to 2nd paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References both applying Full-Kelly,Half-Kelly and also Quarter-Kelly models which similar with my previous Kelly-Criterion model ®γσ, Eng Lian Hu 201417 Kindly refer to 4th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References but enhanced.

To achieve the level of profitable betting, one must develop a correct money management procedure. The aim for a punter is to maximize the winnings and minimize the losses. If the punter is capable of predicting accurate probabilities for each match, the Edward O. Thorp (2006)18 Kindly refer to 6th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References has proven to work effectively in betting. It was named after an American economist John Kelly (1956)19 Kindly refer to 26th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References and originally designed for information transmission. The Kelly criterion is described below:

\[S=(\rho*\sigma-1)/(\sigma-1) \cdots equation-4.3.1\]

Where S = the stake expressed as a fraction of one’s total bankroll, \(\rho\) = probability of an event to take place, \(\sigma\) = odds for an event offered by the bookmaker. Three important properties, mentioned by Hausch and Ziemba (1994) (Efficiency of Racetrack Betting Markets (2008Edition)), arise when using this criterion to determine a proper stake for each bet:

The criterion is known to economists and financial theorists by names such as the geometric mean maximizing portfolio strategy, the growth-optimal strategy, the capital growth criterion, etc. We will now show that Kelly betting will maximize the expected log utility for sports-book betting.

[1] 23.71528

\[K = \frac{(B + 1)p - 1} {B} \cdots equation 4.3.1\]

\[G: = \mathop {\lim }\limits_{N \to \infty } \frac{1/N}{\log}\left( {\frac{{{BR_N}}}{{{BR_0}}}} \right) \cdots equation 4.3.2\]

\[BR_N = (1 + K)^W(1 - K)^L BR_0 \cdots equation 4.3.3\]

Kelly K-value 凯利模式资金管理

## Bootstrapping to get the optimal value
#'@ llply(rEMProbB)

table 4.3.2

In order to get the optimal value, I apply the bootrapping and resampling method.

\[L(\rho) = \prod_{i=1}^{n} (x_{i}|\rho) \cdots equation 4.3.4\]

Now we look at abpve function from a different perspective by considering the observed values \(x1, x2, …, xn\) to be fixed parameters of this function, whereas \(\rho\) will be the function’s variable and allowed to vary freely; this function will be called the likelihood.

4.4 Poisson Ⓜodel

Niko Marttinen (2001)20 Kindly refer to 1th paper in Reference for industry knowdelege and academic research portion for the paper. has enhanced the Dixon and Coles (1996) which are :

From above models, the author has compare the efficiency and the best fit model for scores prediction as below.

figure 4.4.1 : Comparison of various Poison models.

figure 4.4.1 : Comparison of various Poison models.

From figure 4.4.1 above, the author compare the deviance of the models21 Kindly refer to Generalized Linear Models in R, Part 2: Understanding Model Fit in Logistic Regression Output, devianceTest and Use of Deviance Statistics for Comparing Models to learn baout the method of comparison.

Here we introduce the Dixon and Coles (1996) model and its codes. You are freely learning from below links if interest.

table 4.4.1 : Filtered multiple bets placed on same matches.

Due to the soccer matches randomly getting from different leagues, and also not Bernoulli win-lose result but half win-lose etc as we see from above. Besides, there were mixed Pre-Games and also In-Play soccer matches and I filter-up the sample data to be 20009 x 45. I don’t pretend to know the correct answer or the model from firm A. However I take a sample presentation Robert Johnson (2011)22 Kindly refer to 23th paper in 7.4 References from one of consultancy firm which is Dixon-Coles model and omitted the scoring process section.

Here I cannot reverse computing from barely \(\rho_i^{EM}\) without know the \(\lambda_{ij}\) and \(\gamma\) values. Therefore I try to using both Home and Away Scores to simulate and test to get the maximum likelihood \(\rho_i^{EM}\).

\[X_{ij} = pois(\gamma \alpha_{ij} \beta_{ij} ); Y_{ij} = pois(\alpha_{ij} \beta_{ij}) \cdots equation 4.4.1\]

sample…

In order to minimzie the risk, I tried to validate the odds price range invested by firm A.23 As I used to work in AS3388 which always take bets from Starlizard where they only placed bets within the odds price range from 0.70 ~ -0.70. They are not placed bets on all odds price in same edge. The sportbook consulatancy firms will not place same amount of stakes on same edge, lets take example as below :-

We know above edge is same but due to the probability of occurance an event/goal at 0.4 is smaller than 0.64. Here I try to bootstrap/resampling the scores of matches of the dataset and apply maximum likelihood on the poisson model to test the Kelly model and get the mean/likelihood value. Boostrapping the scores and staking model will be falling in the following sections [4.5 Staking Ⓜodel and Ⓜoney Management] and 4.6 Expectation Ⓜaximization and Staking Simulation.

4.5 Staking Ⓜodel and Ⓜoney Ⓜanagement

Martin Spann and Bernd Skiera (2009)24 Kindly refer to 19th paper in 7.4 References applied a basic probability sets on the draw games and also the portion of win and loss. The author simply measured the portion of the draw result with win/loss to get the edge to place a bet. However it made a loss on Italian operator Oddset due to the 25% high vigorish but profitable in 12%. Secondly, the bets placed on fixed odds but not Asian Handicap and also a fixed amount $100.

sample… Geometric Mean

Parimutuel Betting

4.6 Expectation Ⓜaximization and Staking Simulation

sample…

5. Result

5.1 Comparison of the Results

Chapter 4.2 Comparison of Different Feature Sets and Betting Strategies in

Dixon and Pope (2003) apply linear model to compare the efficiency of the odds prices offer by first three largest Firm A, B and C in UK.

5.2 Market Basket

Here I apply the arules and arulesViz packages to analyse the market basket of the bets.

6. Conclusion

6.1 Conclusion

Due to the data-sets I collected just one among all agents among couple sports-bookmakers 4lowin. Here I cannot determine if the sample data among the population…

JA : What skills and academic training (example: college courses) are valuable to sports statisticians?

KW : I would say there are three sets of skills you need to be a successful sports statistician:

  • Quantitative skills - the statistical and mathematical techniques you’ll use to make sense of the data. Most kinds of coursework you’d find in an applied statistics program will be helpful. Regression methods, hypothesis testing, confidence intervals, inference, probability, ANOVA, multivariate analysis, linear and logistic models, clustering, time series, and data mining/machine learning would all be applicable. I’d include in this category designing charts, graphs, and other data visualizations to help present and communicate results.
  • Technical skills - learning one or more statistical software systems such as R/S-PLUS, SAS, SPSS, Stata, Matlab, etc. will give you the tools to apply quantitative skills in practice. Beyond that, the more self-reliant you are at extracting and manipulating your data directly, the more quickly you can explore your data and test ideas. So being adept with the technology you’re likely to encounter will help tremendously. Most of the information you’d be dealing with in sports statistics would be in a database, so learning SQL or another query language is important. In addition, mastering advanced spreadsheet skills such as pivot tables, macros, scripting, and chart customization would be useful.
  • Domain knowledge - truly understanding the sport you want to analyze professionally is critical to being successful. Knowing the rules of the game; studying how front offices operate; finding out how players are recruited, developed, and evaluated; and even just learning the jargon used within the industry will help you integrate into the organization. You’ll come to understand what problems are important to the GM and other decisionmakers, as well as what information is available, how it’s collected, what it means, and what its limitations are. Also, I recommend keeping up with the discussions in your sport’s analytic community so you know about the latest developments and what’s considered the state of the art in the public sphere. One of the great things about being a sports statistician is getting to follow your favorite websites and blogs as a legitimate part of your job!

source : Preparing for a Career as a Sports Statistician: Two Interviews with People in the Field

… … …

6.2 Future Works

Niko Marttinen (2001) has conducted a very detail and useful but also applicable betting system in real life. There has a ordered probit model which shows a high accuracy predictive model compare to his Poisson (Escore) model. Well, the ®γσ, Lian Hu ENG (2016)25 Infact completed in year 2010 has build a weight inflated diagonal poisson model which is more complicated and shophitiscated and later ®γσ, Lian Hu ENG (2014). However there has an automatically and systematically trading system which wrote in VBA + S-Plus + Excel + SQL26 the betting system has stated in his paper. which is very useful as reference. The author use VBA to automac the algorithmic trading while there has no Asian Handicap and Goal Line odds price data to simulate compare to mine. While currently the shinyapps with RStudioConnect can also build an algorithmic trading system. However the session timeout issue27 The connection timeout issue might be a big issue for real time algorithmic trading might need to consider. The shinydashboard example from ｮStudio might probably cope with the issue.

I will be apply Shiny to write a dynamic website to utilise the function as web based apps. I am currently conducting another research on corporate-finance-analysis which is an analysis on the public listed companies and also anonymous companies revenue and profit & loss. You are welcome to refer SHOW ME SHINY and build your own shinyapps.

I will also write as a package to easier load and log.

7. Appendices

7.1 Documenting File Creation

It’s useful to record some information about how your file was created.

[1] “2016-09-16 21:47:41 JST”

7.2 Versions’ Log

7.3 Speech and Blooper

Firstly I do appreciate those who shade me a light on my research. Meanwhile I do happy and learn from the research.

Due to the rmarkdown file has quite some sections and titles, you might expand or collapse the codes by refer to Code Folding and Sections for easier reading.

There are quite some errors when I knit HTML:

7.4 References

Reference for industry knowdelege and academic research portion for the paper.

  1. Creating a Profitable Betting Strategy for Football by Using Statistical Modelling by Niko Marttinen (2006)
  2. What Actually Wins Soccer Matches: Prediction of the 2011-2012 Premier League for Fun and Profit by Jeffrey Alan Logan Snyder (2013)
  3. Odds Modelling and Testing Inefficiency of Sports Bookmakers : Rmodel by ®γσ, Eng Lian Hu (2016)
  4. Apply Kelly-Criterion on English Soccer 2011/12 to 2012/13 by ®γσ, Eng Lian Hu (2014)
  5. The Betting Machine by Martin Belgau Ellefsrød (2013)
  6. The Kelly Criterion in Blackjack Sports Betting, and the Stock Market by Edward Thorp (2016)
  7. Statistical Methodology for Profitable Sports Gambling by Fabián Enrique Moya (2012)
  8. How to apply the Kelly criterion when expected return may be negative? by user1443 (2011)
  9. Money Management Using The Kelly Criterion by Justin Kuepper
  10. Optimal Exchange Betting Strategy For WIN-DRAW-LOSS Markets by Darren O’Shaughnessy (2012)
  11. Kelly criterion with more than two outcomes by David Speyer (2014)
  12. 凯利模式资金管理 by Chung-Han Hsieh (2015)
  13. Optimal Determination of Bookmakers’ Betting Odds: Theory and Tests by John Fingleton & Patrick Waldron (1999)
  14. Optimal Pricing in the Online Betting Market by Maurizio Montone (2015)
  15. Why are Gambling Markets Organised so Differently from Financial Markets? by Steven Levitt (2004)
  16. Forecasting Accuracy and Line Changes in the NFL and College Football Betting Markets by Steven Xu (2013)
  17. The Forecast Ability of the Dispersion of Bookmaker Odds by Kwinten Derave (2013-2014)
  18. The Stocks at Stake: Return and Risk in Socially Responsible Investment by Galema, Plantinga and Scholtens (2008)
  19. A Comparison of the Forecast Accuracy of Prediction Markets, Betting Odds and Tipsters by Martin Spann and Bernd Skiera (2009)
  20. Efficiency of the Market for Racetrack Betting by Donald Hausch, William Ziemba and Mark Rubinstein (1981)
  21. Betting Market Efficient at Premiere Racetracks by Marshall Gramm (2011)
  22. Late Money and Betting Market Efficiency: Evidence from Australia by Marshall Gramm, Nicholas McKinney and Randall Parker (2012)
  23. An introduction to football modelling at Smartodds by Robert Johnson (2011)
  24. The Value of Statistical Forecasts in the UK Association Football Betting Market by Dixon and Pope (2003)
  25. Modelling Association Football Scores and Inefficiencies in the Football Betting Market by Dixon & Coles (1996)
  26. A New Interpretation of Information Rate by John Kelly (1956)
  27. Dynamic Modelling and Prediction of English Football League Matches for Betting by Crowder, Dixon, Ledford and Robinson (2001)